The Batcave is the secret underground headquarters of Batman (formerly also to Nightwing and Robin, while also sometimes for Batgirl), consisting of a subterranean complex beneath his alter-ego Bruce Wayne's residence, containing all of the numerous sophisticated pieces of equipment utilized in his long anti-crime crusade.
What became the Batcave was first discovered by Bruce Wayne on the day of his parents' funeral. As their caskets were being taken to the Wayne Mausoleum, Bruce ran off into the woods. As he kept on running he fell down into a well into a cave with bats, which would inspire him with an idea.
As Bruce grew older and developed the idea to rid Gotham City of the criminals and corrupt, he used the series of the caves underneath the Wayne Manor he fell as a child as his base of operations. Along with Alfred Pennyworth, the two of them built up an underground armory and high tech operations area over the years, with a number of levels along with a secret tunnel entrance that leads from underneath the lake on Wayne Manor's grounds. After Wayne Manor was partially destroyed, Bruce and Alfred installed an elevator leading directly to the Batcave from Bruce's new home, the Glasshouse.